## Osaka Journal of Mathematics

- Osaka J. Math.
- Volume 56, Number 2 (2019), 289-299.

### Reduced contragredient Lie algebras and PC Lie algebras

#### Abstract

Using the theory of standard pentads, we can embed an arbitrary finite-dimensional reductive Lie algebra and its finite-dimensional completely reducible representation into some larger graded Lie algebra. However, it is not easy to find the structure of the ``larger graded Lie algebra'' from the definition in general cases. Under these, the first aim of this paper is to show that the ``larger graded Lie algebra'' is isomorphic to some PC Lie algebra, which are Lie algebras corresponding to special standard pentads called pentads of Cartan type. The second aim is to find the structure of a PC Lie algebra.

#### Article information

**Source**

Osaka J. Math., Volume 56, Number 2 (2019), 289-299.

**Dates**

First available in Project Euclid: 3 April 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.ojm/1554278426

**Mathematical Reviews number (MathSciNet)**

MR3934977

**Zentralblatt MATH identifier**

07080086

**Subjects**

Primary: 17B65: Infinite-dimensional Lie (super)algebras [See also 22E65]

Secondary: 17B67: Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 17B70: Graded Lie (super)algebras

#### Citation

Sasano, Nagatoshi. Reduced contragredient Lie algebras and PC Lie algebras. Osaka J. Math. 56 (2019), no. 2, 289--299. https://projecteuclid.org/euclid.ojm/1554278426